Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4014,2,Mod(1,4014)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4014, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4014.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4014 = 2 \cdot 3^{2} \cdot 223 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4014.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(32.0519513713\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.103354048.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 8x^{4} + 14x^{3} + 13x^{2} - 16x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{6} - 2x^{5} - 8x^{4} + 14x^{3} + 13x^{2} - 16x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} + \nu^{3} - 7\nu^{2} - 6\nu + 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 8\nu^{3} + 6\nu^{2} + 13\nu - 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 8\nu^{3} + 7\nu^{2} + 13\nu - 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{5} - 6\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{5} - 7\beta_{4} + \beta_{3} + 7\beta_{2} + 28\beta _1 + 3 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1.00000 | 0 | 1.00000 | −2.93762 | 0 | 2.50626 | 1.00000 | 0 | −2.93762 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0 | 1.00000 | −2.63291 | 0 | −2.24656 | 1.00000 | 0 | −2.63291 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0 | 1.00000 | −1.03898 | 0 | −0.299718 | 1.00000 | 0 | −1.03898 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0 | 1.00000 | 2.51659 | 0 | 4.97725 | 1.00000 | 0 | 2.51659 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0 | 1.00000 | 2.56494 | 0 | 2.27923 | 1.00000 | 0 | 2.56494 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 0 | 1.00000 | 3.52797 | 0 | 0.783532 | 1.00000 | 0 | 3.52797 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(223\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4014.2.a.u | yes | 6 |
| 3.b | odd | 2 | 1 | 4014.2.a.t | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4014.2.a.t | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 4014.2.a.u | yes | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):
|
\( T_{5}^{6} - 2T_{5}^{5} - 19T_{5}^{4} + 30T_{5}^{3} + 110T_{5}^{2} - 112T_{5} - 183 \)
|
|
\( T_{7}^{6} - 8T_{7}^{5} + 11T_{7}^{4} + 36T_{7}^{3} - 84T_{7}^{2} + 22T_{7} + 15 \)
|
|
\( T_{11}^{6} - 2T_{11}^{5} - 22T_{11}^{4} + 28T_{11}^{3} + 75T_{11}^{2} + 42T_{11} + 7 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots - 183 \)
|
| $7$ |
\( T^{6} - 8 T^{5} + \cdots + 15 \)
|
| $11$ |
\( T^{6} - 2 T^{5} + \cdots + 7 \)
|
| $13$ |
\( T^{6} + 2 T^{5} + \cdots + 1 \)
|
| $17$ |
\( T^{6} - 2 T^{5} + \cdots - 71 \)
|
| $19$ |
\( T^{6} + 2 T^{5} + \cdots + 283 \)
|
| $23$ |
\( T^{6} - 22 T^{5} + \cdots + 1043 \)
|
| $29$ |
\( T^{6} - 8 T^{5} + \cdots - 18015 \)
|
| $31$ |
\( T^{6} + 12 T^{5} + \cdots + 35 \)
|
| $37$ |
\( T^{6} - 76 T^{4} + \cdots + 237 \)
|
| $41$ |
\( T^{6} - 28 T^{5} + \cdots - 16439 \)
|
| $43$ |
\( T^{6} - 14 T^{5} + \cdots + 327 \)
|
| $47$ |
\( T^{6} - 2 T^{5} + \cdots + 2787 \)
|
| $53$ |
\( T^{6} - 26 T^{5} + \cdots - 79803 \)
|
| $59$ |
\( T^{6} - 4 T^{5} + \cdots - 2537 \)
|
| $61$ |
\( T^{6} + 6 T^{5} + \cdots + 78065 \)
|
| $67$ |
\( T^{6} - 18 T^{5} + \cdots + 80067 \)
|
| $71$ |
\( T^{6} - 12 T^{5} + \cdots + 111575 \)
|
| $73$ |
\( T^{6} + 20 T^{5} + \cdots + 19197 \)
|
| $79$ |
\( T^{6} - 12 T^{5} + \cdots + 15 \)
|
| $83$ |
\( T^{6} - 12 T^{5} + \cdots - 1757 \)
|
| $89$ |
\( T^{6} + 2 T^{5} + \cdots - 99563 \)
|
| $97$ |
\( T^{6} + 6 T^{5} + \cdots - 372155 \)
|
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